
This is called the variable projection functional (VPF). Note that the estimation of \(\)  is now a nonlinear least squares problem in \(j\)  dimensions; whereas, the original problem has \(j + k\)  coefficients.
Equation REF  permits an explicit calculation of the cost function \(E()\)  as a function of \(\)  for a given measurement \(y̭\) .
In Section 4 of [1]}, Golub provides the formal proofs required to determine that the Fréchet derivatives over the pseudo-inverse and projection yield the gradient of the cost function. This implies that gradient based nonlinear solvers can be used to find the value of \(\)  which minimizes the cost function \(E()\) .
